Translation invariant valuations on quasi-concave functions

TL;DR

This paper develops a theoretical framework for translation invariant valuations on quasi-concave functions, including a decomposition theorem, a representation formula for homogeneous valuations, and the introduction of Klain's functions for this context.

Contribution

It introduces a homogeneous decomposition theorem and a representation formula for N-homogeneous valuations, along with defining Klain's functions for quasi-concave functions.

Findings

Proved a McMullen-type homogeneous decomposition theorem.

Derived a representation formula for N-homogeneous valuations.

Introduced Klain's functions for valuations on quasi-concave functions.

Abstract

We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations.